Rough Weierstrass functions and dynamical systems: the smoothness of the SBR measure

Peter Imkeller; Olivier  Menoukeu Pamen; Goncalo Dos Reis; Anthony Reveillac

Rough Weierstrass functions and dynamical systems: the smoothness of the SBR measure

Peter Imkeller, Olivier Menoukeu Pamen, Goncalo Dos Reis, Anthony Reveillac

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate Weierstrass functions with roughness parameter γ that are Hölder continuous with coefficient H=logγ/log12. Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough γ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times.

Original language	English
Number of pages	29
Journal	Pure and Applied Functional Analysis
Publication status	Accepted/In press - 21 Apr 2021

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2009.03628Accepted author manuscript, 3.85 MB

https://arxiv.org/abs/2009.03628

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title = "Rough Weierstrass functions and dynamical systems: the smoothness of the SBR measure",

abstract = "We investigate Weierstrass functions with roughness parameter γ that are H{\"o}lder continuous with coefficient H=logγ/log12. Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough γ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times. ",

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AU - Menoukeu Pamen, Olivier

AU - Dos Reis, Goncalo

AU - Reveillac, Anthony

PY - 2021/4/21

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N2 - We investigate Weierstrass functions with roughness parameter γ that are Hölder continuous with coefficient H=logγ/log12. Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough γ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times.

AB - We investigate Weierstrass functions with roughness parameter γ that are Hölder continuous with coefficient H=logγ/log12. Analytical access is provided by an embedding into a dynamical system related to the baker transform where the graphs of the functions are identified as their global attractors. They possess stable manifolds hosting Sinai-Bowen-Ruelle (SBR) measures. We systematically exploit a telescoping property of associated measures to give an alternative proof of the absolute continuity of the SBR measure for large enough γ with square-integrable density. Telescoping allows a macroscopic argument using the transversality of the flow related to the mapping describing the stable manifold. The smoothness of the SBR measure can be used to compute the Hausdorff dimension of the graphs of the original Weierstrass functions and investigate their local times.

M3 - Article

JO - Pure and Applied Functional Analysis

JF - Pure and Applied Functional Analysis

SN - 2189-3756

ER -

Rough Weierstrass functions and dynamical systems: the smoothness of the SBR measure

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